Hello all, I've been lurking here for a bit, reading and searching on all kinds of interesting/scary plumbing stories. Thanks speedball and all the rest for all your free knowledge you bestow upon us mere mortals.

Here goes- it's long so you'll have all the details!

Situation:
Big house, 6500sf+, gas hot water down stairs (brand new 40gal GE) with Grundfos water circulator pump, electric 20 gal WH upstairs. Upstairs is great, with great water pressure, etc. we'll be concentrating downstairs.

Problem:
Water Circulator
-If the Grundfos water circulator is off, you CAN'T get hot water to the house. You actually can, but I've only gotten it to the kitchen sink (20 feet from the HW heater, & it's not really hot- just warm)- and that took 5 minutes. If the circulator is on, the you can get hot water to the far leg of the house in 1.5 minutes. I thought it was the circulator pump blocking the hot water when it was not on, so therefore you HAD to have it on to get hot water. The pump just died, and I put a straight pipe where the pump used to be. It acts the same as when the pump was in place, and not on- I can't get hot water to the house. I tried at the farthest leg (the farthest leg as far as I can tell is the 'shower', about 100' away) for 7 minutes, then gave up.
-I think all the piping is 3/4"

Other oddities:
-When the pump was working, and I had hot water, hot water would only last maybe 10 minutes. There was no way to take a bath at all.
-These actions are the same with 2 different gas 40 gal water heaters- I just installed a new GE 40 gallon gas to replace an older gas water heater- so the water heater is not the problem.
-if I turn on the pump, and get hot water to the shower, then go turn off the pump while the shower is going, it'll go cold again. The pump must be on for hot water.
-there is a water softener outside of the house even beyond the 'shower.' Maybe that is the longest run, or maybe that is where the water comes in from the city. It was 'wired around' by just bypassing it by soldering the in/out tubes together (as far as I can see) many years ago.
-the house is made for a pump, it has a return line, and it's a slab house so I don't think it was retrofitted.
-I don't know where the 'loop back' pipe is (the pipe that joins the hot/cold sides for the circulator) I assume it's near the shower (what we'll call the furthest leg- but the water softener might actually be the furthest leg)
-Downstairs water pressure stinks. Always has. Not unbearable, but nothing like upstairs.


Things I don't know:
-where does the water from the city come in? Near/at the hot water heater? Near the water softener? Somewhere else? It must T and go upstairs somewhere before all this downstairs mess.
-don't know where the 'loopback' pipe is- looked under the sink at the shower. Could it be/been at the water softener?

Things I want to know:
-why can't I run hot water w/o a pump?

My conclusions (guesses):
While I'm in the shower, I philosophize why this acts so strange. I have not come up with much.
-the 'loop back' pipe is stuck open- as far as I can tell, they are supposed to contain a bi-metallic system to open/close the valve, and that has something to do with it. Maybe it's just drawing the 'hot' water (actually cold) from the cold return line, besides the hot water line- maybe due to water flowing easier from the cold line compared to the hot water line
-my other thought is that there are trolls that live in the walls that control valves to my hot water.. . as you can see, I have not got too far with this.

Is the pump located at the H/W tank, or is it remotely located at a fixture elsewhere?

It is (was) located right next to the HW tank.

It sounds like the check valves weren't installed at all, as opposed to being incorrectly installed.

This is the order that the shutoffs, check valves and tee's should have been installed.

A) Coming out of the wall or dropping out of the ceiling, there should have been a shut-off on the cold water side

B) Immediately after that should have been a check valve with the arrow pointing towards direction of flow into the cold inlet of the H/W tank.

C) There should be a tee after the first valve, with the outlet of the tee going into the cold water inlet of the tank.

D) The center outlet of the tee should have another check valve in it with the direction of flow going towards the cold water inlet of the tank.

E) The pump, with the direction of flow going into the H/W tank, should be installed into the inlet of the check valve.

F) And finally, there should be a shut-off installed on the return line before the return line enters the inlet of the pump.